Small roots, low elements, and the weak order in Coxeter groups

In this article we provide a new finite class of elements in any Coxeter system (W,S)(W,S) called low elements  . They are defined from Brink and Howlett's small roots, which are strongly linked to the automatic structure of (W,S)(W,S). Our first main result is to show that they form a Garside shadow in (W,S)(W,S), i.e., they contain S and are closed under join (for the right weak order) and by taking suffixes. These low elements are the key to prove that all finitely generated Artin–Tits groups have a finite Garside family. This result was announced in a note with P. Dehornoy in Comptes rendus mathématiques [9] in which the present article was referred to under the following working title: Monotonicity of dominance-depth on root systems and applications.The proof is based on a fundamental property enjoyed by small roots and which is our second main result; the set of small root is bipodal.For a natural number n, we define similarly n-low elements from n-small roots and conjecture that the set of n-small roots is bipodal, implying the set of n-low elements is a Garside shadow; we prove this conjecture for affine Coxeter groups and Coxeter groups whose graph is labelled by 3 and ∞. To prove the latter, we extend the root poset on positive roots to a weak order on the root system and define a Bruhat order on the root system, and study the paths in those orders in order to establish a criterion to prove bipodality involving only finite dihedral reflection subgroups.